Total Families

To guarantee the termination and completeness of the solving of equality constraints, we need to impose rather draconian restrictions on the instances of type synonym families. More specifically, to achieve both termination and completeness, we need the Strong Termination Condition and if we settle for termination alone (accepting to be incomplete for some rather exotic programs), we need the Relaxed (Termination) Condition as defined in ​Type Checking with Open Type Functions. The Strong Termination Condition corresponds closely to the conditions imposed on functional dependencies. The Relaxed Condition is somewhat more liberal, but still does not permit, for example, nested applications of type families in the right-hand side of a type instance. For many practically attractive uses of type families, where the the system is actually terminating, this is still to restrictive.

The family (:+) meets the Relaxed Condition, but not the Strong Termination Condition. However, (:*) meets not even the Relaxed Condition. Nevertheless, we would expect that families defined by structural recursion should not compromise termination. Another somewhat irritating aspect of this example is that we would like these definitions to be closed, but they are open in their present form.

Unfortunately, the two instances are overlapping and there is no means by which we can disambiguate the overlap by using the same textual ordering as that which we are used to from value-level functions.

Defining total families

To enable textual disambiguation of overlapping instances, we declare the equalities together (by transferring GADT syntax to type synonyms):

type TypeEq s t where
TypeEq s s = TTrue
TypeEq s t = TFalse

TypeEq is a standard type family, but by virtue of being total (i.e., exhaustive) it is also closed. Further equalities cannot define it any further.

In contrast to TypeEq, (:+) and (:*) is not total without an extra equality. We take another idea from value-level function definitions and implicitly complete each of these definitions by a final catch all equality. So, for (:+), we assume a final